Preheat the oven to 350 degrees F (175 degrees C). Spread 1 tablespoon olive oil on a baking sheet; set aside.

Heat 1 tablespoon olive oil in a skillet over medium heat. Stir in shallots; cook until translucent, about 3 minutes. Stir in garlic, thyme, salt, and pepper; cook and stir for 2 minutes. Transfer shallot mixture to a bowl. Mix in goat cheese, dates, and basil; stir until well-combined.

Cut a 1-inch long slit with a sharp knife into the thick side of each chicken breast. Work your fingers into the slit and expand the slit to form a pocket.

With your fingers or a spoon, stuff each chicken breast with about 1/4 cup goat cheese mixture. Wipe off any cheese mixture from the outside of chicken breast. Wrap each breast in a slice of prosciutto so that the pocket opening is covered. Place chicken breasts, seam-sides down, onto the prepared baking sheet.

Bake in the preheated oven for 20 minutes. Flip chicken and continue baking until chicken is no longer pink and prosciutto is browned and crisp, about 20 minutes more.